Question

Dr. V.R. Bencivenga and Dr. Stephen Donald Economics 329 HOMEWORK QUESTION - Belief in global warming In a large city a certain (unknown) proportion of adults believe that global warming is a liberal hoax designed to hurt oil companies. You plan to take a survey of n adults to "estimate" this proportion. You want to be able to say that your estimate is within three percentage points (i.e., within 0.03) of the true proportion, with 0.95 probability. What is the smallest sample size that guarantees that you would obtain a satisfactory estimate (an estimate that's within 0.03 of the true proportion, with probability 0.95)? Provide an answer using an approximation based on the Central Limit Theorem a. 752 b. 2135 c. 4269 d. 1068

Answer 1

Answer: d. 1068

Step-by-step explanation:

Let p be the proportion of adults believe that global warming is a liberal hoax designed to hurt oil companies.

Given : You plan to take a survey of n adults to "estimate" this proportion. You want to be able to say that your estimate is within three percentage points (i.e., within 0.03) of the true proportion, with 0.95 probability.

Margin of error : E= 0.03

We know that the z-value for 95% confidence = $z_c=1.96$

Since the prior proportion of adults believe that global warming is a liberal hoax designed to hurt oil companies.  

We assume p = 0.5

Then by Central Limit Theorem , the required sample size would be :

$n=p(1-p)(\dfrac{z_{c}}{E})^2$

$\Rightarrow\ n=0.5(1-0.5)(\dfrac{1.96}{0.03})^2$

Simply , we get

$n=1067.11111111\approx1068$  [Rounded to the next whole number.]

Hence, the smallest sample size that guarantees that you would obtain a satisfactory estimate  =1068